Computational law is the branch of legal informatics concerned with the automation of legal reasoning. What distinguishes Computational Law systems from other instances of legal technology is their autonomy, i.e. the ability to answer legal questions without additional input from human legal experts. While there are many possible applications of Computational Law, the primary focus of work in the field today is compliance management, i.e. the development and deployment of computer systems capable of assessing, facilitating, or enforcing compliance with rules and regulations. Some systems of this sort already exist. TurboTax is a good example. And the potential is particularly significant now due to recent technological advances – including the prevalence of the Internet in human interaction and the proliferation of embedded computer systems (such as smart phones, self-driving cars, and robots). There are also applications that do not involve governmental laws. The regulations can just as well be the terms of contracts (e.g. delivery schedules, insurance covenants, real estate transactions, financial agreements). They can be the policies of corporations (e.g. constraints on travel, expenditure reporting, pricing rules). They can even be the rules of games (embodied in computer game playing systems). == History == Speculation about potential benefits to legal practice through applying methods from computational science and AI research to automate parts of the law date back at least to the middle 1940s. Further, AI and law and computational law do not seem easily separable, as perhaps most of AI research focusing on the law and its automation appears to utilize computational methods. The forms that speculation took are multiple and not all related in ways to readily show closeness to one another. This history will sketch them as they were, attempting to show relationships where they can be found to have existed. By 1949, a minor academic field aiming to incorporate electronic and computational methods to legal problems had been founded by American legal scholars, called jurimetrics. Though broadly said to be concerned with the application of the "methods of science" to the law, these methods were actually of a quite specifically defined scope. Jurimetrics was to be "concerned with such matters as the quantitative analysis of judicial behavior, the application of communication and information theory to legal expression, the use of mathematical logic in law, the retrieval of legal data by electronic and mechanical means, and the formulation of a calculus of legal predictability". These interests led in 1959 to the founding a journal, Modern Uses of Logic in Law, as a forum wherein articles would be published about the applications of techniques such as mathematical logic, engineering, statistics, etc. to the legal study and development. In 1966, this Journal was renamed as Jurimetrics. Today, however, the journal and meaning of jurimetrics seems to have broadened far beyond what would fit under the areas of applications of computers and computational methods to law. Today the journal not only publishes articles on such practices as found in computational law, but has broadened jurimetrical concerns to mean also things like the use of social science in law or the "policy implications [of] and legislative and administrative control of science". Independently in 1958, at the Conference for the Mechanization of Thought held at the National Physical Laboratory in Teddington, Middlesex, UK, the French jurist Lucien Mehl presented a paper both on the benefits of using computational methods for law and on the potential means to use such methods to automate law for a discussion that included AI luminaries like Marvin Minsky. Mehl believed that the law could by automated by two basic distinct, though not wholly separable, types of machine. These were the "documentary or information machine", which would provide the legal researcher quick access to relevant case precedents and legal scholarship, and the "consultation machine", which would be "capable of answering any question put to it over a vast field of law". The latter type of machine would be able to basically do much of a lawyer's job by simply giving the "exact answer to a [legal] problem put to it". By 1970, Mehl's first type of machine, one that would be able to retrieve information, had been accomplished but there seems to have been little consideration of further fruitful intersections between AI and legal research. There were, however, still hopes that computers could model the lawyer's thought processes through computational methods and then apply that capacity to solve legal problems, thus automating and improving legal services via increased efficiency as well as shedding light on the nature of legal reasoning. By the late 1970s, computer science and the affordability of computer technology had progressed enough that the retrieval of "legal data by electronic and mechanical means" had been achieved by machines fitting Mehl's first type and were in common use in American law firms. During this time, research focused on improving the goals of the early 1970s occurred, with programs like Taxman being worked on in order to both bring useful computer technology into the law as practical aids and to help specify the exact nature of legal concepts. Nonetheless, progress on the second type of machine, one that would more fully automate the law, remained relatively inert. Research into machines that could answer questions in the way that Mehl's consultation machine would picked up somewhat in the late 1970s and 1980s. A 1979 convention in Swansea, Wales marked the first international effort solely to focus upon applying artificial intelligence research to legal problems in order to "consider how computers can be used to discover and apply the legal norms embedded within the written sources of the law". Considerable progress on the development of the second type of machine was made in the following decade, with the development of a variety of expert systems. According to Thorne McCarty, "these systems all have the following characteristics: They do backward chaining inference from a specified goal; they ask questions to elicit information from the user; and they produce a suggested answer along with a trace of the supporting legal rules." According to Prakken and Sartor the representation of the British Nationality Act as a logic program, which introduced this approach, was "hugely influential for the development of computational representations of legislation, showing how logic programming enables intuitively appealing representations that can be directly deployed to generate automatic inferences". In 2021, this work received the Inaugural CodeX Prize as "one of the first and best-known works in computational law, and one of the most widely cited papers in the field." In a 1988 review of Anne Gardner's book An Artificial Intelligence Approach to Legal Reasoning (1987), the Harvard academic legal scholar and computer scientist Edwina Rissland wrote that "She plays, in part, the role of pioneer; artificial intelligence ("AI") techniques have not yet been widely applied to perform legal tasks. Therefore, Gardner, and this review, first describe and define the field, then demonstrate a working model in the domain of contract offer and acceptance." Eight years after the Swansea conference had passed, and still AI and law researchers merely trying to delineate the field could be described by their own kind as "pioneer[s]". In the 1990s and early 2000s more progress occurred. Computational research generated insights for law. The First International Conference on AI and the Law occurred in 1987, but it is in the 1990s and 2000s that the biannual conference began to build up steam and to delve more deeply into the issues involved with work intersecting computational methods, AI, and law. Classes began to be taught to undergraduates on the uses of computational methods to automating, understanding, and obeying the law. Further, by 2005, a team largely composed of Stanford computer scientists from the Stanford Logic group had devoted themselves to studying the uses of computational techniques to the law. Computational methods in fact advanced enough that members of the legal profession began in the 2000s to both analyze, predict and worry about the potential future of computational law and a new academic field of computational legal studies seems to be now well established. As insight into what such scholars see in the law's future due in part to computational law, here is quote from a recent conference about the "New Normal" for the legal profession: "Over the last 5 years, in the fallout of the Great Recession, the legal profession has entered the era of the New Normal. Notably, a series of forces related to technological change, globalization, and the pressure to do more with less (in both corpo
Hierarchical navigable small world
Hierarchical navigable small world (HNSW) is an algorithm for approximate nearest neighbor search. It is used to find items that are similar to a query item in a large collection, without comparing the query with every item one by one. The algorithm is commonly used for searching vector data. In these systems, an item such as a document, image, song, or user profile is represented by a list of numbers called a vector. Items with similar vectors are treated as similar according to the model that produced the vectors. HNSW provides a way to search these vectors quickly, especially in large datasets. HNSW stores vectors in a graph. Each vector is a node, and links connect it to some nearby vectors. The graph has several layers: upper layers contain fewer nodes and act like a rough map, while the bottom layer contains all nodes and gives a more detailed view. A search starts in an upper layer, follows links toward nodes that are closer to the query, and then repeats the process in lower layers until it finds a set of likely nearest neighbors. == Background == The nearest neighbor search problem asks which items in a dataset are closest to a query item. A direct search can compare the query with every item in the dataset, but this becomes slow when the dataset is large. Exact search methods based on spatial trees, such as the k-d tree and R-tree, can also become less effective for high-dimensional data, a problem often associated with the curse of dimensionality. Approximate nearest neighbor methods trade some exactness for speed or lower resource use. Instead of always guaranteeing the exact closest item, they try to return close items quickly. Other approximate methods include locality-sensitive hashing and product quantization. HNSW builds on research into small-world networks and navigable graphs. In a small-world graph, most nodes can be reached from other nodes through a short chain of links. In a navigable graph, a search procedure can use local information to move toward a target. Jon Kleinberg's work on navigation in small-world networks is an important example of this research area. Later work studied ways to add links that make graphs easier to navigate greedily. The HNSW algorithm extends earlier navigable small world methods for similarity search by adding a hierarchy of graph layers. This hierarchy helps the algorithm find a good region of the graph before doing a more detailed search in the bottom layer. == Algorithm == HNSW is based on a proximity graph. In this graph, nearby vectors are connected by edges. The algorithm uses these edges to move through the dataset, rather than scanning every vector. The graph is hierarchical. Every vector appears in the bottom layer. Some vectors are also placed in higher layers, with fewer vectors appearing as the layers go upward. The upper layers allow long-range movement across the dataset, while the lower layers allow a more detailed search near promising candidates. A typical search proceeds as follows: The search begins from an entry point in the highest layer. At each step, the algorithm looks at neighboring nodes and moves to a neighbor that is closer to the query. When it cannot find a closer neighbor in that layer, it moves down to the next layer. In the bottom layer, it explores a wider set of candidate nodes and returns the nearest candidates found. This search strategy is often described as greedy navigation. The algorithm repeatedly chooses locally better nodes, using the graph structure to approach the query point. == Construction and parameters == The HNSW graph is built incrementally. When a new vector is inserted, the algorithm assigns it a maximum layer, searches for nearby existing nodes, and connects the new node to selected neighbors in each layer where it appears. Implementations usually expose parameters that control the trade-off between speed, accuracy, memory use, and construction time. A higher number of graph connections can improve recall but requires more memory. A larger search candidate list can improve accuracy but makes queries slower. A larger construction candidate list can improve the quality of the graph but makes index building slower. Because HNSW is approximate, its results are not always identical to a full exact search. Its practical performance depends on the dataset, distance measure, implementation, and parameter settings. Benchmarking studies have found HNSW-based libraries to be strong performers among approximate nearest neighbor methods, although worst-case performance can differ from performance on common benchmark datasets. == Use in vector search systems == HNSW is used as an index in systems that store and search high-dimensional vectors. These systems include vector databases, search engines, and database extensions. Typical uses include semantic search, recommender systems, image similarity search, and retrieval-augmented generation. Several software projects implement or support HNSW. Libraries include hnswlib, which is associated with the original HNSW authors, and FAISS. Database and search systems that document HNSW support include Apache Lucene, Chroma, ClickHouse, DuckDB, MariaDB, Milvus, pgvector, Qdrant, and Redis.
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Xuedong Huang
Xuedong David Huang (born October 20, 1962) is a Chinese-American computer scientist and technology executive who has made contributions to spoken language processing and artificial intelligence, including Azure AI Services. He is Zoom's chief technology officer after serving as Microsoft's Technical Fellow and Azure AI Chief Technology Officer for 30 years. Huang is a strong advocate of AI for Accessibility, and AI for Cultural Heritage. == Education == Huang received his PhD from the University of Edinburgh in 1989 (sponsored by the British ORS and Edinburgh University Scholarship), his MS from Tsinghua University in 1984, and BS from Hunan University in 1982. == Career == After receiving his PhD in 1989, Huang joined Carnegie Mellon University and worked with Raj Reddy and Kai-Fu Lee on speech recognition. At CMU, he directed the Sphinx-II speech system research which achieved the best performance in every category of DARPA's 1992 benchmarking. Microsoft Research recruited him to found and lead Microsoft's spoken language initiatives in 1993. His co-authored book Spoken Language Processing and his Historical speech recognition review succinctly summarize several generations of spoken language research. As Microsoft's Mr. Speech for three decades, Huang has been instrumental in creating Microsoft's Speech Application Programming Interface (SAPI), shipping Microsoft Speech Server, and modernizing spoken language and integrative AI services via Azure AI, which not only enables millions of 3rd party customers but also powers up Microsoft's Windows, Office, Teams, and Azure OpenAI Services. Huang helped Microsoft and Azure Cognitive Services achieve multiple industry's first human parity milestones on the following open research tasks: transcribing conversational speech, machine translation, conversational QnA, and computer vision image captioning. Huang has made significant contributions to the software and AI industry through his executive leadership and his scientific publications, owning more than 170 US patents and impacting billions through Azure AI enabled products and services. In 2016, Wired magazine named him one of 25 Geniuses. In 2021, Azure AI was named the winner of InfoWorld's Technology of the Year Award. Huang was awarded the Allen Newell research excellence medal in 1992, and IEEE Speech Processing Best Paper in 1993. He was recognized as an IEEE Fellow by Institute of Electrical and Electronics Engineers in 2000, named ACM Fellow by Association for Computing Machinery in 2017, and a member of Washington State Academy of Sciences. Huang received 2022 Asian American Corporate Leadership Award, and IEEE Amar Bose Industrial Leader Award. In 2023, he was elected a member of the US National Academy of Engineering (NAE), and a member of the American Academy of Arts and Sciences.
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Neural operators
Neural operators are a class of deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial neural networks, marking a departure from the typical focus on learning mappings between finite-dimensional Euclidean spaces or finite sets. Neural operators directly learn operators between function spaces; they can receive input functions, and the output function can be evaluated at any discretization. The primary application of neural operators is in learning surrogate maps for the solution operators of partial differential equations (PDEs), which are critical tools in modeling the natural environment. Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers. Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data, and the geosciences. In particular, they have been applied to learning stress-strain fields in materials, classifying complex data like spatial transcriptomics, predicting multiphase flow in porous media, and carbon dioxide migration simulations. Finally, the operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces, and subsumes these settings as special cases when limited to a fixed input resolution. == Operator learning == Understanding and mapping relationships between function spaces has many applications in engineering and the sciences. In particular, one can cast the problem of solving partial differential equations as identifying a map between function spaces, such as from an initial condition to a time-evolved state. In other PDEs this map takes an input coefficient function and outputs a solution function. Operator learning is a machine learning paradigm to learn solution operators mapping the input function to the output function . Using traditional machine learning methods, addressing this problem would involve discretizing the infinite-dimensional input and output function spaces into finite-dimensional grids and applying standard learning models, such as neural networks. This approach reduces the operator learning to finite-dimensional function learning and has some limitations, such as generalizing to discretizations beyond the grid used in training. The primary properties of neural operators that differentiate them from traditional neural networks is discretization invariance and discretization convergence. Unlike conventional neural networks, which are fixed on the discretization of training data, neural operators can adapt to various discretizations without re-training. This property improves the robustness and applicability of neural operators in different scenarios, providing consistent performance across different resolutions and grids. == Definition and formulation == Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are composed of alternating linear maps and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear integral operators on function spaces and point-wise non-linearities. Using an analogous architecture to finite-dimensional neural networks, similar universal approximation theorems have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a compact set. Neural operators seek to approximate some operator G : A → U {\displaystyle {\mathcal {G}}:{\mathcal {A}}\to {\mathcal {U}}} between function spaces A {\displaystyle {\mathcal {A}}} and U {\displaystyle {\mathcal {U}}} by building a parametric map G ϕ : A → U {\displaystyle {\mathcal {G}}_{\phi }:{\mathcal {A}}\to {\mathcal {U}}} . Such parametric maps G ϕ {\displaystyle {\mathcal {G}}_{\phi }} can generally be defined in the form G ϕ := Q ∘ σ ( W T + K T + b T ) ∘ ⋯ ∘ σ ( W 1 + K 1 + b 1 ) ∘ P , {\displaystyle {\mathcal {G}}_{\phi }:={\mathcal {Q}}\circ \sigma (W_{T}+{\mathcal {K}}_{T}+b_{T})\circ \cdots \circ \sigma (W_{1}+{\mathcal {K}}_{1}+b_{1})\circ {\mathcal {P}},} where P , Q {\displaystyle {\mathcal {P}},{\mathcal {Q}}} are the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function to the output dimension) operators, respectively. These operators act pointwise on functions and are typically parametrized as multilayer perceptrons. σ {\displaystyle \sigma } is a pointwise nonlinearity, such as a rectified linear unit (ReLU), or a Gaussian error linear unit (GeLU). Each layer t = 1 , … , T {\displaystyle t=1,\dots ,T} has a respective local operator W t {\displaystyle W_{t}} (usually parameterized by a pointwise neural network), a kernel integral operator K t {\displaystyle {\mathcal {K}}_{t}} , and a bias function b t {\displaystyle b_{t}} . Given some intermediate functional representation v t {\displaystyle v_{t}} with domain D {\displaystyle D} in the t {\displaystyle t} -th hidden layer, a kernel integral operator K ϕ {\displaystyle {\mathcal {K}}_{\phi }} is defined as ( K ϕ v t ) ( x ) := ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x):=\int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy,} where the kernel κ ϕ {\displaystyle \kappa _{\phi }} is a learnable implicit neural network, parametrized by ϕ {\displaystyle \phi } . In practice, one is often given the input function to the neural operator at a specific resolution. For instance, consider the setting where one is given the evaluation of v t {\displaystyle v_{t}} at n {\displaystyle n} points { y j } j n {\displaystyle \{y_{j}\}_{j}^{n}} . Borrowing from Nyström integral approximation methods such as Riemann sum integration and Gaussian quadrature, the above integral operation can be computed as follows: ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y ≈ ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j , {\displaystyle \int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy\approx \sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}},} where Δ y j {\displaystyle \Delta _{y_{j}}} is the sub-area volume or quadrature weight associated to the point y j {\displaystyle y_{j}} . Thus, a simplified layer can be computed as v t + 1 ( x ) ≈ σ ( ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j + W t ( v t ( y j ) ) + b t ( x ) ) . {\displaystyle v_{t+1}(x)\approx \sigma \left(\sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}}+W_{t}(v_{t}(y_{j}))+b_{t}(x)\right).} The above approximation, along with parametrizing κ ϕ {\displaystyle \kappa _{\phi }} as an implicit neural network, results in the graph neural operator (GNO). There have been various parameterizations of neural operators for different applications. These typically differ in their parameterization of κ {\displaystyle \kappa } . The most popular instantiation is the Fourier neural operator (FNO). FNO takes κ ϕ ( x , y , v t ( x ) , v t ( y ) ) := κ ϕ ( x − y ) {\displaystyle \kappa _{\phi }(x,y,v_{t}(x),v_{t}(y)):=\kappa _{\phi }(x-y)} and by applying the convolution theorem, arrives at the following parameterization of the kernel integral operator: ( K ϕ v t ) ( x ) = F − 1 ( R ϕ ⋅ ( F v t ) ) ( x ) , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x)={\mathcal {F}}^{-1}(R_{\phi }\cdot ({\mathcal {F}}v_{t}))(x),} where F {\displaystyle {\mathcal {F}}} represents the Fourier transform and R ϕ {\displaystyle R_{\phi }} represents the Fourier transform of some periodic function κ ϕ {\displaystyle \kappa _{\phi }} . That is, FNO parameterizes the kernel integration directly in Fourier space, using a prescribed number of Fourier modes. When the grid at which the input function is presented is uniform, the Fourier transform can be approximated using the discrete Fourier transform (DFT) with frequencies below some specified threshold. The discrete Fourier transform can be computed using a fast Fourier transform (FFT) implementation. == Training == Training neural operators is similar to the training process for a traditional neural network. Neural operators are typically trained in some Lp norm or Sobolev norm. In particular, for a dataset { ( a i , u i ) } i = 1 N {\displaystyle \{(a_{i},u_{i})\}_{i=1}^{N}} of size N {\displaystyle N} , neural operators minimize (a discretization of) L U ( { ( a i , u i ) } i = 1 N ) := ∑ i = 1 N ‖ u i − G θ ( a i ) ‖ U 2 {\displaystyle {\mathcal {L}}_{\mathca
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